Introduction | Object Counts | Still-lifes | P2 Oscillators | P4 Oscillators | Billiard table oscillators | Spaceships | Wick-stretchers

The B2/S2 rule is similar to Life, except both birth and survival occur on exactly 2 living neighbors. This rule has the following interesting properties:

- The birth-on-2 rule makes any leading surface containing a domino expand
forever at a speed of
*c*, making it impossible for anything to catch up and stop it. This makes possible a very simple spaceship, as well as infinitely many dirty puffers. This also makes this rule very difficult to work with, since almost every pattern expands without limit. - Due to the birth-on-2 rule, pseudo-objects are not possible.
- Survival-on-2, combined with lack of birth on 3-6, make possible solid walls that are totally impervious to assault from either side. In essence, they are immovable objects, except at their ends. If such walls bend around corners, their inside corners are also impervious. Four such walls surrounding an enclosed space make the space totally contained, and useful for creating billiard table oscillators of arbitrarily high periods.
- Still-lifes are possible, but they are all rare. They must all be surrounded by L trominos, and if sufficiently large, may also contain enclosed chains of walls (see previous section).
- Since the birth and survival rules are identical, a cell does not
participate in its subsequent state. I.e., no information is transferred
directly from a cell and its child. Information can only be transferred
from a cell to its subsequent generation by moving to an alternate cell
and coming back later. This principle attaches parity to information,
and it is no surprise that most oscillators have periods of
2, or sometimes
4. The one common
spaceship mechanism has a period of 1 and
a velocity of
*c*, demonstrating the same parity principle. The many dirty puffers derived from it all appear to have even periods. - Unlike Life, where "cool" broth exteriors usually muddle around and occasionally expand, the birth-on-2 rule causes almost every broth to expand in every orthogonal direction. In terms of object synthesis, the practical result of this is extremely difficult to collide gliders together to leave only small usable sparks. It is even extremely difficult to even destroy an object with gliders. For this reason, all syntheses on this page also include object destructor mechanisms, if available.

Computer searches have counted still-lifes up to 30 bits and period 2 oscillators up to 17 bits.

Numbers in **bold face** have been confirmed by computer search. Other
given numbers are believed to be complete, but have not yet been verified.
Lists with large numbers of objects that have not yet been counted are shown
with "many".

Status of object lists on sub-pages is shown by background color:

- White (or light blue) indicates small complete object list; may have syntheses for some.
- Red indicates complete object count; may have syntheses for some.
- Grey indicates objects not counted; may have syntheses for some.

Bits |
2 | 3 |
4 | 5 |
6 | 7 |
8 | 9 |
10 | 11 |
12 | 13 |
14 | 15 |
16 | 17 |
18 | 19 |
20 | 21 |
22 | 23 |
24 | 25 |
26 | 27 |
29 | 29 |
30 |

Still-lifes |
0 | 0 | 0 |
0 | 0 | 0 |
0 | 0 |
0 | 0 | 1 |
0 | 0 |
0 | 0 | 0 |
1 |
0 |
0 | 0 | 0 |
0 | 4 |
0 | 0 | 3 |
0 | 0 |
13 |

P2 oscillators |
1 | 1 | 0 |
0 | 3 | 3 |
5 | 2 | 15 |
26 | 56 |
103 | 250 |
478 | 1129 |
many | |||||||||||||

P4 oscillators |
0 | 1 | 2 | 0
| 1 | 2 | 4 | 0 | 2 | 8 | 21 | 13 | many | ||||||||||||||||

Spaceships |
0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 7 | many |

Still-lifes are extremely rare in this rule. They are all highly artificial, and must employ extensive frameworks to induct all unwanted births on the outside. Since survival can only happen on 2, the only components permitted are L trominos and circular chains of bits (typically rounded rectangles, although at larger sizes, concave lakes are also possible). Since births can only occur on 2 neighbors, and such chains are bounded by empty cells with more than 2 everywhere except at outside corners, they act as stable immovable objects everywhere except at their outside corners. (As a side-effect, this permits construction of totally safe billiard table enclosures, since no matter what happens on the inside, the outside is safe from harm.)

Stable arrangements can be made of rectangular arrays of L trominos
containing at least 2 in each dimension. Furthermore, one rectangular corner can
be removed, yielding an L-shaped arrangement, as long as the two arms have width
2 or more. Thus, it is possible to create still-lifes of any population
3(*xy*+*xj*+*iy*) where *x*,*y*≥2 and
*i*,*j*≥0. This simplifies to 3*n* for *n* in 4, 6,
or any number 8 and larger.

A single pond can be stabilized by 16 trominos, allowing still-lifes of
populations 56+3(3*i*+2*j*) where *i**j*≥0,
or 56+3*n* where *n*≥0.

Finally, a hollow box with 6 or more bits on the side can be stabilized by
3 L trominos on each corner, yielding still-lifes of populations 60+2*n*.
If an additional L tromino is added on an inside corner, this yields still-lifes
of populations 63+2*n*.

Thus, still-lifes are possible with populations 12, 18, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 56, 57, 60, and all sizes 62 and larger.

The following table shows all still-lifes up to 30 bits, plus a sampling of all other known sizes up to 65 bits. (Note that the number above each object indicates its population; none can be synthesized from gliders, and individual pattern files are not provided.)

Although not generally considered a still-life, the empty field (i.e. death) technically fulfills all the criteria, and it warrants being mentioned for several reasons. First, it is extremely rare for any patterns to totally die out. Second, unlike any of the other still lifes, it can be synthesized from gliders. Third, there are only three collisions of two gliders that don't expand infinitely, and one of them results in death.

These are the 56 period 2 oscillators up to 11 bits, plus two larger ones for which syntheses are known (and that are both infinitely extensible).

Blinker (2.1) [6] | Yoyo (3.1) [8] | 6.1 [x] | 6.2 [x] | 6.3 [x] | 7.1 [x] | 7.2 [x] | 7.3 [x] | 8.1 [x] | 8.2 [x] |

8.3 [x] | 8.4 [x] | 8.5 [x] | 9.1 [x] | 9.2 [x] | 10.1 [x] | 10.2 [x] | 10.3 [x] | 10.4 [x] | 10.5 [x] |

10.6 [x] | 10.7 [x] | Lines (10.8) [2] | 10.9 [x] | 10.10 [x] | 10.11 [x] | 10.12 [x] | 10.13 [x] | 10.14 [x] | 10.15 [x] |

11.1 [x] | 11.2 [x] | 11.3 [x] | 11.4 [x] | 11.5 [x] | 11.6 [x] | 11.7 [x] | 11.8 [x] | 11.9 [x] | 11.10 [x] |

11.11 [x] | 11.12 [x] | 11.13 [x] | 11.14 [x] | 11.15 [x] | 11.16 [x] | 11.17 [x] | 11.18 [x] | 11.19 [x] | 11.20 [x] |

11.21 [x] | 11.22 [x] | 11.23 [x] | 11.24 [x] | 11.25 [x] | 11.26 [x] | Three lines [4] | Long lines [4] |

Other than the numerous period 2 oscillators, and the eccentric large billiard tables, the only other known oscillators are three small period 4 pulsators, plus their trivial extensions. Ones marked * are infinitely extensible. All up to 10 bits are shown, plus a few larger ones.

Since still-lifes rely on heavily inducting the outsides of interiors that are inherently unstable, the same mechanisms used to construct them can also construct a variety of billiard table oscillators. A few small ones work with only L tromino stabilizations, but most work safely within a totally enclosed box. The period 5 oscillator shown below is infinitely extensible; a domino placed 5 or more bits away from both edges of a channel is a bi-directional period 5 glider gun that shoots gliders into the walls at both ends. (None of these can be synthesized from gliders, and individual pattern files are not provided.)

The last example shown is not really an oscillator at all. It was originally believed to be an oscillator with a period in excess of 20000, but was finally analyzed by Golly, using its HashLife algorithm, and the interior stablizes into a single blinker after 2,878,904 generations. It is a good illustration of the total robustness of billiard table interiors in this rule.

There is one basic spaceship, that has a period of 1, and moves with a
velocity of *c*. Most variations of it produce incredibly dirty puffers
with even periods, although there are a few that are clean spaceships and
wick stretchers. Spaceships of any period 2^{n} can be
constructed by extending the examples below.

Two other spaceships have been found with velocities slower than *c*
but these are very rare in this rule.

P4 Escorted domino Spaceship [x] | P8 Escorted domino Spaceship [x] |
c/2
Glider 21193 [x] |

P2 Spaceship [x] | P2 Escorted domino Spaceship [x] | |

P1 Glider; Glider 230 [4] |
c/3 Glider 9588 [x] |

The basic *c* spaceship can cleanly extend lines perpendicular to its
direction of travel. This makes construction of wick-stretchers easy.

The tube-stretchers are special wick-stretchers that extrude an
ever-expanding tube similar to a billiard table. Anything can be safely put
inside, it cannot escape; the bottom and sides are impervious, and
the top is receding at *c*, so nothing can catch up to it. The rightmost example
shows how a single extra bit can forment total chaos, but in this case, the
chaos is contained within a constrained area.

Two-line wick- stretcher [x] | Blinkers on line wick- stretcher [x] | Empty tube- stretcher [x] | Glider- filled tube- stretcher [x] | Chaos- filled tube- stretcher [x] |

**Other rules:**
B3/S23 (Conway's Life),
Multi-colored Life,
B2/S2 (2/2 Life),
B34/S34 (3/4 Life),
Niemiec's Rules.

**See also:**
definitions,
structure,
search methodologies,
other rules,
news,
credits,
links,
site map,
search,
expanded search,
search help,
downloads.

Copyright © 1997, 1998, 1999, 2013, 2014 by Mark. D. Niemiec.
All rights reserved.

This page was last updated on
*2015-02-19*.